Cascade transversal-filter phase-compensation network

ABSTRACT

A phase-compensation network, capable of modifying the phase response of a filter or network while leaving unchanged the amplitude response, comprising a cascaded combination of simple transversal filters, each of which comprises a delay line; at least one tapped weighted element whose input is connected to the delay line; and a signal summer whose input is connected to the outputs of the weighted elements. The elements of each simple transversal filter correspond to the values of the Bessel function of fixed argument and for successive integral indices of the order, including the zeroth order, only significant values of positive and negative indices of the order being used, the element corresponding to the zeroth order being in the center of its specific transversal filter. The output of one transversal filter constitutes the input to the next succeeding filter in the cascade, each transversal filter corresponding to one of a set of fixed arguments of a Bessel function of the first kind, the set of fixed arguments being obtained from the coefficients of a phase function when expressed in Fourier series form.

United States Patent Byram et al.

CASCADE TRANSVERSAL-FILTER PHASE-COMPENSATION NETWORK Inventors: GeorgeW. Byram; Jeffrey M.

Speiser, both of San Diego, Calif.

The United States of America as represented by the Secretary of theNavy, Washington, DC.

Filed: Oct. 15, 1973 Appl. No.: 406,720

[73] Assignee:

US. Cl 333/70 T, 333/28 R Int. Cl H03h 7/28, H03h 7/30, H04b 3/04 Fieldof Search 333/70 T, 28 R, 18;

References Cited UNITED STATES PATENTS 12/1971 Perreault 333/70 TX [5 7ABSTRACT A phase-compensation network, capable of modifying the phaseresponse of a filter or network while leaving unchanged the amplituderesponse, comprising a cascaded combination of simple transversalfilters, each of which comprises a delay line; at least one tappedweighted element whose input is connected to the delay line; and asignal summer whose input is connected to the outputs of the weightedelements. The elements of each simple transversal filter correspond tothe values of the Bessel function of fixed argument and for successiveintegral indices of the order, including the zeroth order, onlysignificant values of positive and negative indices of the order beingused, the element corresponding to the zeroth order being in the centerof its specific transversal filter. The output of one transversal filterconstitutes the input to the next succeeding filter in the cascade, eachtransversal filter corresponding to one of a set of fixed arguments of aBessel function of the first kind, the set of fixed arguments beingobtained from the coefficients of a phase function when expressed inFourier series form.

3 Claims, 5 Drawing Figures IvPur I 941-22 94(0) A 1:) Q 2 j I 104- 6!)104(0) 104 (I) ,7; p. 071) @25 r100 i 2 j J 774/20 DEL/1y Luvs a iOur-p07 Avon/EA EM50DIMEA/7 01: A pmspcampsusnrmf Main/0R1.

CASCADE TRANSVERSAL-FILTER PHASE-COMPENSATION NETWORK STATEMENT OFGOVERNMENT INTEREST The invention described herein may be manufacturedand used by or for the Government of the United States of America forgovernmental purposes without the payment of any royalities thereon ortherefor.

BACKGROUND OF THE INVENTION This invention relates to a general linearfilter with transfer function HQ) e P i.e., a general all-pass orphase-compensation network, using a cascade combination of simpletransversal filters.

Such a network may be used to modify the phase response of an existingnetwork or filter, while leaving the amplitude response unchanged. Sincethe phase response of a filter, amplifier, or other linear system iscritical in many signal processing applications, the invention has wideutility.

Arbitrary phase-compensation functions may be implemented in a verysimple manner. A small total number of delay line taps is required,reducing the effect of the spurious dispersion which would otherwise beintroduced by the taps themselves interacting with the propagating wave.Since each of the cascaded trasversal filters requires no more than 20taps (and usually a very much smaller number of taps), the fabricationof the individual filters is relatively straightforward. Not only arethe individual filters easy to build, but the computation of therequired tap weights is very simple.

A small number of filters may be combined in various combinations toprovide a large family of phase compensation functions.

DESCRIPTION OF THE PRIOR ART The prior art techniques used in the areaof a general all-pass or phase-compensation network generally fall intothree categories: (a) lumped network synthesis; (b) dispersive delaylines; and (c) single transversal filters.

The design of a lumped network to have a prescribed phase response anduniform amplitude response is extremely difficult, and both thedifficulty of design and the component sensitivity grow rapidly with thetimebandwidth product of the desired impulse response or transferfunction.

While dispersive delay lines with high timebandwidth products have beenbuilt, it is difficult to generate an arbitrary dispersion function bythis method. The primary utility of this method is for obtaining filtersmatched to linear FM or quadratic FM signals.

The single transversal filter provides a more flexible method ofsynthesis than the lumped network or dispersive filter, in general onlyrequiring a tapped delay line with 2TW independent taps, where TW is thetime bandwith product of the desired impulse response. If thetime-bandwidth product is sufficiently large, however, then this methodof synthesis also becomes difficult to use for several reasons: (1)Tapped delay lines of sufficient length may not exist. For example, inthe case of acoustic surface-wave delay lines, the length is limited bythe size of the available crystals. (2) Unwanted attenuation anddispersion due to energy extracted from the wave in propagating past alarge number of taps. (3) Secondary signal generation effects in anacoustic surface wave device. The acroustic wave may perturb the inputvoltage appearing across the launch transducer, thus launching asecondary acoustic wave.

This invention relates to a phase-compensation network, capable ofmodifying the phase response of a filter or network while leavingunchanged the amplitude response, comprising: a cascaded combination ofsimple transversal filters, each of which in turn comprises: a delayline; at least one tapped weighted element whose input is connected tothe delay line; and a signal summer whose input is connected to theoutputs of the weighted elements.

The elements of each simple transversal filter correspond to the valuesof the Bessel function of fixed arugment and for successive integralindices of the order, including the zeroth order. Only significantvalues of positive and negative indices of the order are used, theelement corresponding to the zeroth order being in the center of itsspecific transversal filter. The elements corresponding to order plusland minus 1, plus 2 and minus 2, and higher orders with theirnegatives, are symmetrically disposed about the central element, thepolarity of two symmetrically disposed elements being the same if theindex of the order is even, and unlike if the order is odd.

The output of one transversal filter constitutes the input to the nextsucceeding filter in the cascade. Each transversal filter correspondingtoone of a set of fixed arguments of a Bessel function of the firstkind, the set of fixed arguments being obtained from the coefficients ofa phase function when expressed in Fourier series form.

OBJECTS OF THE INVENTION An object of the invention is to provide aphasecompensation network which may be implemented in a very simplemanner, since it requires use of a comparatively small number of taps.

Another object of the invention is to provide a phasecompensationnetwork in which computation of the required tap weights, for eachfilter, is very simple.

Yet another object of the invention is to provide a filter which may becombined with other filters to provide a family of phase compensationfunctions.

Other objects, advantages and novel features of the invention willbecome apparent from the following detailed description of theinvention, when considered in conjunction with the accompanyingdrawings, wherein:

BRIEF DESCRIPTION OF THE DRAWINGS FIG. 1 is a graph of an arbitraryphase function.

FIG. 2 is a graph of a phase function with the pure delay term removed.

FIG. 3 is a graph showing the magnitude of the Fourier coefficients as afunction of the index of the Fourier coefficients.

FIG. 4 is a block diagram of a phase-compensation network.

FIG. 5 is a block diagram of another embodiment of a phase-compensationnetwork having relatively few taps.

3 I 4 I But it may be-shown that wmm'w' DESCRIPTION OF THE PREFERRED thB EMBODIMENTS g J (z where 1,, denotes the m essel func tion of thefirst kind, Before discussing specific embodiments, it should and g,whenever n is not a multiple of k. prove useful to explain the theorybehind the invention. 5 Th k term i di t h at l i l of k k b i aninteger, as well as m, g equals J (z that is, for Let the desired filtertransfer function be I) every integer value f m and k, Jm(zk) 8E0, e' PFirst, suppose that the filter may be realized (in whenever n is not amultiple of principle) by a Single transversal filter with Spacing Themk term has a minus sign, otherwise the reversal and lmpulse' responseof the tap weights would comprise the Bessel function.

h(t) =2 h 8(tkd),'

Ir Each 2,, is less than 1r in absolute value. But for such moderatearguments, the Bessel functions fall off rapidly in magnitude as theorder is increased, resulting in very few taps being needed for each ofthe cascaded transversal filters. H f f r h(t)dt-'=2 h e**"' Referringnow to FIG. 1, the phase response shown k H in curve 10 of FIG. 1 is thedesign objective, ignoring which i s aperiodic function of frequency,with period the realization delays, which introduce a lineal Phase d- Sih(;) i l, H(-f) ]-[*(f) where h t ,trend. The design is involved withcurve 20 of FIG. 2, isk denotes complex conjugation. This in turnrequires but the curve eventually realized resembles curve 10 in thatoff) be an odd function of f (modulo 2n). Since FIG. 1, because of thepresence of the realization delay.

where 8(1) is the Dirac delta function. From which it follows that thecorresponding transfer function in Foul5 rier transform form is (b isodd and periodic, it may be explained in a Fourier A comparison of thecurve 10 shown in FIG. 1 with sine series: that, curve 20, shown in FIG.2, reveals that both curves are similar in shape, but that curve 20 isrotated f =2 2,, sin 21rkfd. with respect to curve 10, in a manner suchthat both end points are on the d) axis.

The equation for curve 20 in FIG. 2 is For many phase functions ofinterest, it suffices to I use a very few terms in the sine seriesexpansion, say

The product on the right side of the above equation corresponds to thecascade combination of N time invariant linear filters. It will be shownhereinbelow that each may be realized as a transversal filter using a Ki i x/2 L .flfma.r- It IS iconvenlent to mul ply the tapped delay linewith a very small number of taps, and and Slde the equatlon h parametera it will be shown how to calculate the required tap lgnatmg the maxlmumPhase devlatlonweights The equation for (f) is the standard Besselsequence The transfer function of the kth cascade filter is I-I U) forthe curve Show in and y be obtained e sin 21rkfd. Since this is aperiodic function of frefrom a Standard mathematlcal handbookquency,with period d", it may be represented in a In the equation for #0) itwill be noted that y Odd complex Fourier series: terms are involved.Therefore, the tap weights would a correspond only to the odd orderterms. Effectively,

H f 2 nr FIG. 3 shows only the magnitude of the terms, with the minussigns indicating when the term is negative. As shown, the magnitude forthe 7 term is negligible.

where d d Hk(f)e df Then for the a 1r case, the coefficients become:

8/1r 2.544, 8/91r 0.283, 8/251r 0.104, 8/4971' V 0.052, -1 l I d f 4. ek e df The above coefficients are used in the embodiment 30 shown inFIG. 4.

For the a 'n'/4 case, the coefficients become:

It is noted that H (f) is the transfer function of a transversal filterwith impulse response 8. /3 0 8/ l "I 8/l967r0.0l3 2 M 0 m Theimmediately above coefficients are used in the eml odiment 80.

For the above values of J,,(x), the following two tables of values maybe determined, as a function of the index p.

Referring mow to FIG. 4, therein is shown a phasecompensation network30, capable of modifying the phase response of a filter or network whileleaving unchanged the amplitude response, comprising a cascadedcombination of simple transversal filters, 40, 50, 60 or 70, each ofwhich comprises a delay line, 42, 52, 62 or 72; at least one tappedweighted element, 44, 54, 64 or 74, whose input is connected to thedelay line; and a signal summer, 46, 56, 66 or 76, whose input isconnected to the outputs of the weighted elements.

The elements, for example, 44( -5), 44(0), 44(5), of each simpletransversal filter, for example, 40, correspond to the values of theBessel function of fixed argument and for successive integral indices ofthe order, including the zeroth order, only significant values ofpositive and negative indices of the order being used. The element 44(0)corresponding to the zeroth order is in the center of its specifictransversal filter 40, the elements corrsponding to orders plus l andminus 1, 44(1) and 44(1), plus 2 and minus 2, 44(2) and 44(-2), andhigher orders with their negatives, being symmetrically disposed aboutthe central element, the polarity of two symmetrically disposed elementsbeing the same if the index of the order is even, and unlike if theorder is odd.

The output 48, 58 or 68, of one transversal filter, 40, S or 60,constitutes the input to the next succeeding filter, each transversalfilter corresponding to one of a set of fixed arguments of a Besselfunction of the first kind, the set of fixed arguments being obtainedfrom the coefficients of a phase function when expressed in Fourierseries form. The output 78 of the last signal summer 76 constitutes theoutput of the phasecompensation network 30.

In summary, in the phase-compensation network 30 in FIG. 4 or 80 in FIG.5 the phase function expanded in a Fourier series may take the form ofwhich, considering only significant values, may be truncated to Thetransfer function of the kth cascade filter is where g J,,,(Z, where'Jdenotes the mth Bessel function of the first kind, and g" 0 whenever nis not a multiple of k.

In FIG. 5 is shown a phase-compensation network 80 wherein the cascadedcombination of transversal filters 90, 100 and 110, comprises threedelay lines 92, 102 and 112, each having a series of weighted elements,94, 104 and 114, connected to it. One series of elements, 94(2), 94(0),94(2), weighted according to the Bessel function J,,(0.638); anotherseries of elements, l04(l), 104(0) and 104(1), being weighted accordingto J,,(0.07l); and the third series of elements, l14(l), 114(0) and114(1), being weighted according to J,,(0.025).

With respect to variations in embodiments of the invention, instead ofsimply truncating the Fourier sine series for the phase function d (f),the phase function may be smoothed prior to taking the truncatedexpansion. Similarly a Cesaro approximating sum may be used. Thisrequires only a minor modification of the design procedure, changing thearguments of the Bessel function used to select the taps weights.

The cascaded transversal filters may be implemented using acousticsurface wave delay lines, torsional magnetic delay lines, or any othertapped delay line with low dispersion, and lightly coupled,low-deflection taps.

Obviously many modifications and variations of the present invention arepossible in the light of the above teachings. It is therefore to beunderstood that within the scope of the appended claims the inventionmay be practiced otherwise than as specifically described.

What is claimed is:

l. A phase-compensation network, capable of modi fying the phaseresponse of a filter or network while leaving unchanged the amplituderesponse, comprising:

a cascaded combination of simple transversal filters, each comprising: adelay line; at least one tapped weighted element whose input isconnected to the delay line;

the elements of each simple transversal filter corresponding to thevalues of the Bessel function of fixed argument and for successiveintegral indices of the order, including the zeroth order, onlysignificant values of positive and negative indices of the order beingused, the element correspond ing to the zeroth order being in the centerof its specific transversal filter, the elements corresponding to ordersplus 1 and minus 1, plus 2 and minus 2, and higher orders with theirnegatives, being symmetrically disposed about the central element, thepolarity of two symmetrically disposed elements being the same if theindex of the order is even, and unlike if the order is odd; and

a signal summer whose input is connected to the outputs of the weightedelements; and wherein the output of one transversal filter constitutesthe input to the next succeeding filter in the cascade,

each transversal filter corresponding to one of a set of fixed argumentsof a Bessel function of the first kind, the set of fixed arguments beingobtained from the coefficients of phase function when expressed inFourier series form.

2. The phase-compensation network according to claim 1, wherein:

the form of the phase function expanded in a Fourier series is 'd (f) =214,, sin Zn-Afd,

which, considering only significant values, may be truncated to (f) i Msin 211. Y k=1 and the transfer function of the kth cascade filter is

1. A phase-compensation network, capable of modifying the phase responseof a filter or network while leaving unchanged the amplitude response,comprising: a cascaded combination of simple transversal filters, eachcomprising: a delay line; at least one tapped weighted element whoseinput is connected to the delay line; the elements of each simpletransversal filter corresponding to the values of the Bessel function offixed argument and for successive integral indices of the order,including the zeroth order, only significant values of positive andnegative indices of the order being used, the element corresponding tothe zeroth order being in the center of its specific transversal filter,the elements corresponding to orders plus 1 and minus 1, plus 2 andminus 2, and higher orders with their negatives, being symmetricallydisposed about the central element, the polarity of two symmetricallydisposed elements being the same if the index of the order is even, andunlike if the order is odd; and a signal summer whose input is connectedto the outputs of the weighted elements; and wherein the output of onetransversal filter constitutes the input to the next succeeding filterin the cascade, each transversal filter corresponding to one of a set offixed arguments of a Bessel function of the first kind, the set of fixedarguments being obtained from the coefficients of phase function whenexpressed in Fourier series form.
 2. The phase-compensation networkaccording to claim 1, wherein: the form of the phase function expandedin a Fourier series is
 3. The phase-compensation network according toclaim 2, wherein the cascaded combination of transversal filterscomprises three delay lines, each having a series of weighted elementsconnected it; one series of elements being weighted according to theBessel function Jp(0.638); another series of elements being weightedaccording to Jp(0.071); and the third series of elements being weightedaccording to Jp(0.025).